Properties

Label 3024.2.bf.d
Level $3024$
Weight $2$
Character orbit 3024.bf
Analytic conductor $24.147$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(1711,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1711");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.bf (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{5} + ( - \zeta_{6} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{5} + ( - \zeta_{6} + 3) q^{7} + (6 \zeta_{6} - 3) q^{11} + ( - 3 \zeta_{6} - 3) q^{13} + ( - \zeta_{6} - 1) q^{17} + \zeta_{6} q^{19} + (2 \zeta_{6} - 1) q^{23} + 2 q^{25} + 9 \zeta_{6} q^{29} + 8 \zeta_{6} q^{31} + ( - 5 \zeta_{6} + 1) q^{35} + 11 \zeta_{6} q^{37} + (\zeta_{6} + 1) q^{41} + ( - 5 \zeta_{6} + 10) q^{43} + ( - 12 \zeta_{6} + 12) q^{47} + ( - 5 \zeta_{6} + 8) q^{49} + (3 \zeta_{6} - 3) q^{53} + 9 q^{55} - 12 \zeta_{6} q^{59} + (9 \zeta_{6} - 9) q^{65} + (2 \zeta_{6} - 4) q^{67} + ( - 4 \zeta_{6} + 2) q^{71} + ( - 7 \zeta_{6} - 7) q^{73} + (15 \zeta_{6} - 3) q^{77} + (6 \zeta_{6} + 6) q^{79} + 9 \zeta_{6} q^{83} + (3 \zeta_{6} - 3) q^{85} + ( - 3 \zeta_{6} + 6) q^{89} + ( - 3 \zeta_{6} - 12) q^{91} + ( - \zeta_{6} + 2) q^{95} + (\zeta_{6} - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{7} - 9 q^{13} - 3 q^{17} + q^{19} + 4 q^{25} + 9 q^{29} + 8 q^{31} - 3 q^{35} + 11 q^{37} + 3 q^{41} + 15 q^{43} + 12 q^{47} + 11 q^{49} - 3 q^{53} + 18 q^{55} - 12 q^{59} - 9 q^{65} - 6 q^{67} - 21 q^{73} + 9 q^{77} + 18 q^{79} + 9 q^{83} - 3 q^{85} + 9 q^{89} - 27 q^{91} + 3 q^{95} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\) \(-1\) \(\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1711.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.73205i 0 2.50000 0.866025i 0 0 0
2287.1 0 0 0 1.73205i 0 2.50000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.bf.d 2
3.b odd 2 1 1008.2.bf.d yes 2
4.b odd 2 1 3024.2.bf.a 2
7.d odd 6 1 3024.2.cz.d 2
9.c even 3 1 3024.2.cz.c 2
9.d odd 6 1 1008.2.cz.b yes 2
12.b even 2 1 1008.2.bf.a 2
21.g even 6 1 1008.2.cz.c yes 2
28.f even 6 1 3024.2.cz.c 2
36.f odd 6 1 3024.2.cz.d 2
36.h even 6 1 1008.2.cz.c yes 2
63.k odd 6 1 3024.2.bf.a 2
63.s even 6 1 1008.2.bf.a 2
84.j odd 6 1 1008.2.cz.b yes 2
252.n even 6 1 inner 3024.2.bf.d 2
252.bn odd 6 1 1008.2.bf.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.a 2 12.b even 2 1
1008.2.bf.a 2 63.s even 6 1
1008.2.bf.d yes 2 3.b odd 2 1
1008.2.bf.d yes 2 252.bn odd 6 1
1008.2.cz.b yes 2 9.d odd 6 1
1008.2.cz.b yes 2 84.j odd 6 1
1008.2.cz.c yes 2 21.g even 6 1
1008.2.cz.c yes 2 36.h even 6 1
3024.2.bf.a 2 4.b odd 2 1
3024.2.bf.a 2 63.k odd 6 1
3024.2.bf.d 2 1.a even 1 1 trivial
3024.2.bf.d 2 252.n even 6 1 inner
3024.2.cz.c 2 9.c even 3 1
3024.2.cz.c 2 28.f even 6 1
3024.2.cz.d 2 7.d odd 6 1
3024.2.cz.d 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$43$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$79$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
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